Two-Person Game on Graphs
Dissertation zur Erlangung des Doktorgrades an der Fakult¨at f¨ur Mathematik
der Universit¨at Bielefeld
vorgelegt von Anush Khachatryan
Erster Gutachter: Prof. Dr. Michael Baake
Introduction v
1 Preliminaries 1
1.1 Basic Definitions . . . 1 1.2 Graph Coloring . . . 3 1.3 Games on Graphs . . . 8 2 The Circular Two-Person Game on Graphs 11 2.1 The Circular Game Chromatic Number of Complete Graphs . . 13 2.2 The Circular Game Chromatic Number of Cycles . . . 15 2.3 The Circular Game Chromatic Number of Complete
Multipar-tite Graphs . . . 17 2.3.1 The Circular Game Chromatic Number of Complete
Bi-partite Graphs without a Perfect Matching . . . 20 2.4 Circular Game Chromatic Number of Planar
Graphs . . . 26 2.4.1 The Activation Strategy for the Circular Game . . . 26 2.5 The Circular Game Chromatic Number of Cactuses . . . 31 3 The Two-Person Game on Weighted Graphs 35 3.1 The Game Chromatic Number of Weighted Complete Graphs . 36 3.2 The Game Chromatic Number of Weighted Complete
Multipar-tite Graphs . . . 45 3.3 The Game Chromatic Number of Weighted Cycles . . . 65 3.4 Construction of Graphs with γ(G, w) < γ(G) . . . 70
3.5 The Game Chromatic Number of Weighted
Trees . . . 73
3.6 The Game Chromatic Number of Weighted Planar Graphs . . . 76
4 The General Asymmetric Game on Graphs 83 4.1 The General Asymmetric Game Chromatic Number of Cycles . 84 4.2 The General Asymmetric Game Chromatic Number of Com-plete Multipartite Graphs . . . 93
4.3 The General Asymmetric Game Chromatic Number of Forests . 101 A Appendix 115 A.1 (k, d)-Coloring and r-Interval Coloring of Graphs . . . 115
A.2 On the Game Chromatic Number of Trees . . . 116
A.3 The Asymmetric Game for the Class of Forests . . . 116
A.3.1 Upper Bounds . . . 116
A.3.2 Lower Bounds . . . 118
First of all, I wish to express my gratitude to my supervisor Prof. Dr. Hans-Georg Carstens for his support and guidance. I would like to thank him for sharing his knowledge, many helpful suggestions and most notably for the academic freedom. I wish to thank Prof. Dr. Baake and Dr. Huck who became my supervisors after Prof. Dr. Cartens’ death on the 28th January, 2012.
Further, I am greatly indebted to the University of Bielefeld and the
Rek-toratsstipendium for financial support which gave me the opportunity to work
on this thesis.
I would like to thank Panagiotis Zacharopoulos for many stimulating con-versations and active interest in the development of this thesis. In particular, I am very thankful for his moral support and patience. Furthermore, I thank my mother for her permanent support and belief in me since the very beginning. Finally, but not less important, thanks to my friends for their motivation and patience; in particular for many helpful suggestions for writing the introduc-tion of this thesis.
A graph G = (V, E) is an ordered pair of disjoint sets V and E such that E is a subset of the set V
2 of unordered pairs of V . V denotes the set of vertices and
E denotes the set of edges of G. A key concept of graph theory is the theory of graph coloring which has its roots in the famous four color conjecture raised by Francis Guthrie 1852. In order to color the regions of a map, he proposed that at most four colors are required so that neighboring regions are assigned different colors. A vertex-coloring of a graph G = (V, E) is a mapping c : V → S such that c(a) 6= c(b) whenever (a, b) ∈ E. The elements of the set S = {1, ..., k} are called the available colors. The chromatic number χ(G) of a graph G is the smallest integer k such that G is colorable.
A modern field of graph coloring is the two-person game on graphs, which goes back to the work of Bodlaender in [3], 1991. Bodlaender defined the two-person game on graphs as follows: Let G = (V, E) be a graph and C a given set of colors. The two players Alice and Bob take turns assigning the vertices of Gcolors from C such that adjacent vertices are assigned different colors. Alice starts the game. She wins, if all vertices of G are colored. Otherwise, if there exists at least one vertex that cannot be colored with a color from C, Bob wins. The game chromatic number of G, denoted by γ(G), is the least cardinality of C, such that there exists a winning strategy for Alice. Obviously, it holds
χ(G)≤ γ(G) ≤ |V |.
The aim of this thesis is to generalize Bodlaender’s idea and to establish three extensions of the two-person game on graphs.
1. The circular two-person game on graphs (Chapter 2) vii
2. The two-person game on weighted graphs (Chapter 3) 3. The general asymmetric game on graphs (Chapter 4)
The first extension is based on the idea of the circular coloring of graphs, introduced by Zhu in [22] (see definition 1.2.1) as an equivalent definition to Vince’s star chromatic number, [20]. By bringing together the two areas the two
person game and the circular coloring, we work out the circular two-person game
in chapter 2 (The Circular Two-Person Game on Graphs) . While in Bodlaender’s person game the given colors are natural numbers, in the circular two-person game Alice and Bob assign to the vertices of a graph unit length arcs on a circle Crwith circumference r ∈ R+. Note that the arcs of adjacent vertices
are not allowed to intersect in a feasible circular coloring. We define the circular
game chromatic number γc(G) of a graph G as the infimum of those r for which
there exists a winning strategy for Alice on Cr. While γ(G) is a natural number,
for the new parameter it holds γc(G)∈ R+. We in particular show that
γc(G) ≤ 2∆(G),
where ∆(G) is the maximum degree of G. The main difficulty in working out a winning strategy for Alice is that one must figure out Bob’s optimal strategy in terms of the arcs he destroys on Cr, that is, they cannot be assigned to any
vertices. Thus, the new game sheds some new light on the respective strategies of the players.
A continuative aim of chapter 2 is to discuss the new parameter for some relevant classes of graphs. We work out winning strategies for Alice for the class of complete graphs and cycles in sections 2.1 and 2.2, respectively. More-over, in section 2.3 we prove for a complete multipartite graph Ks1,...,sn on n
independent sets S1, ..., Sn with |Si| = si, that γc(Ks1,...,sn) ≤ 3n − 2. In 2.3.1
we look more closely at the circular game chromatic number of a complete bipartite graph Ks1,s2 minus a perfect matching M and establish that
γc(Ks1,s2 − M) ≥ γc(Ks1,s2).
Afterwards, we turn our attention to the well known activation strategy, introduced by Kierstead in [11]. A lot of strong results have been won using
Kierstead’s activation strategy, which is a winning strategy for Alice for the
marking game, a simplified version of Bodlaender’s two-person game (see page
8). In [26] Zhu pointed out the notion of the marking game and proved that the marking game number of a graph G, denoted by colg(G), provides an upper
bound of the game chromatic number such that it holds γ(G)≤ colg(G).
We prove that the same conclusion can be drawn for the circular version of the game such that for every graph G it holds
γc(G)≤ colcg(G),
where we define colc
g(G)as the circular marking game number of G. In particular,
we generalize the activation strategy for the case of circular coloring and show that the circular game chromatic number is bounded by a parameter which we define as the circular rank of a graph G. In particular, we use this result and prove for the upper bound for planar graphs that
colc
g(G)≤ 34.
Finally, we turn our attention to the class of cactuses where we use tech-niques by Sidorowicz in [19]. We prove that colc
g(G)is bounded by 8.
Independently Zhu and Lin also dealt with the circular game chromatic number in [27].
Our second extension of the two-person game is the subject of chapter 3
(The Two-Person Game on Weighted Graphs). We consider a vertex-weighted graph
which is a triple (G, w) = (V, E, w) where w is a mapping w : V → N that assigns to every vertex x ∈ V a positive integer w(x) ≥ 1. A coloring of vertex-weighted graphs requires the additional condition c(x) ≥ w(x), where
c : V → N is the coloring mapping. Thus, we define the game chromatic number
of weighted graphs, denoted by γ(G, w), as the smallest amount of given colors,
such that there exists a winning strategy for Alice for the coloring game on a weighted graph (G, w). Obviously, if the mapping w takes the constant value 1, it holds that γ(G) = γ(G, w).
The consideration of the two-person game on weighted graphs provides a generalization of Bodlaender’s two-person game because for a weighted graph (G, w) = (V, E, w) it is assumed that w(v) ≥ 1 for all v ∈ V . The difficulty in carrying out the two-person game on weighted graphs is that during the whole game Alice and Bob have to take not only the structure of the graph but also the vertex-weights into account. In particular, they have to make sure that the color c(v) of a vertex v ∈ V is equal or greater than its weight w(v), while figuring out their optimal strategies.
It is well known and obvious that for the chromatic number of a graph G it holds that χ(G) ≤ χ(G, w). One may conjecture that the same conclusion can be drawn for the two-person game, such that γ(G) ≤ γ(G, w) because w(x) ≥ 1 for all x ∈ V . However, in section 3.4 we work out a surprising result and construct graphs with the property γ(G, w) < γ(G). Furthermore, we characterize this class of graphs in proposition 3.4.3.
Moreover, we wish to investigate the new parameter for some relevant classes of graphs considering certain distributions of vertex-weighs.
In section 3.1 the game chromatic number of a weighted complete graph (Kn, w) on n vertices is investigated. After determining γ(Kn, w) for all
val-ues of w, we establish the relation between the required number of colors for a feasible coloring and wmax(Kn, w)which is the maximum vertex-weight of
the graph. Furthermore, we analyze the new parameter for a complete graph under the assumption w : V → {k, l} where k, l ∈ N and k 6= l.
In the next section we turn our attention to the class of complete multi-partite graphs relating to certain distributions of vertex-weights. We give a lower as well as an upper bound for the entire class. In addition we restrict the weight function such that w(Si)∈ {k, l} for k, l ∈ N and k 6= l, while assuming
that vertices belonging to the same independent set achieve the same weight. In particular, Alice’s strategy has to be adapted for the cases si ≥ 3 and si ≥ 4
for all i ∈ {1, ..., n}. Afterwards, this idea is being generalized such that we work under the assumption that w(Si)6= w(Sj)for all i, j ∈ {1, ..., n} and
ver-tices belonging to the same independent set achieve the same weight. Finally, we drop the assumption that vertices contained in the same independent set have the same weight and analyze the game for the case that w : V → {k, l}
where w(Si)∈ {k, l} for k 6= l.
Afterwards, in section 3.3 we proceed with the game chromatic number of weighted cycles. First, we give a lower bound as well as an upper bound for all values of w and give conditions when they are attained. An interesting result is obtained, if we consider (k1, ..., km)-alternating-weighted cycles, where
for each two vertices xiand xj with distance m it holds w(xi) = w(xj).
In section 3.5 we analyze the game on weighted trees applying an algo-rithm of Faigle, Kern, Kierstead and Trotter in [14]. They estimated the max-imum number of colored adjacent vertices of an uncolored vertex during the game by 3. In particular, for determining the upper bound of γ(T, w), where (T, w)is a weighted tree, we need to observe the weights of the leaves, as well as the interior-vertices of a tree.
Section 3.6 deals with a generalization of Kierstead’s activation strategy by admitting vertex-weights. In particular, we show that the game chromatic number of a weighted graph is bounded by a parameter which we call weighted
rank and give an upper bound of the game chromatic number for weighted
pla-nar graphs. The main difficulty in contrast to the ordipla-nary two-person game is that for a vertex x ∈ V a neighbor y ∈ N(x) with w(y) > w(x) is not necessarily a threat for x. The following example demonstrates this. Let {y1, y2, y3} ∈ N(x)
with w(x) = 10, w(y1) = 5, w(y2) = 11and w(y3) = 20. Vertex y2 with weight
11seems to be harmless for x, because w(x) = 10 and y2has to be colored with
a color greater or equal 11. However, if Bob colors y1 with the color 10 and y2
with the color 11, then x is being attacked twice. On the contrary, x cannot be attacked by y3 since |N(x)| = 3.
x,10 y1,5
y2,11 y3,20
The subject of chapter 4 (General Asymmetric Games on Graphs) is a variation of Bodlaender’s two-person game going back to the work of Kierstead. In [17] he introduced the asymmetric game on graphs. The basic difference to the
ordinary game is that the players color several vertices in a row instead of one vertex each time they take turns; in particular, Alice colors a and Bob b vertices for a, b ∈ N and a, b ≥ 1 (see page 9). Further, a turn is not being completed as soon as either every vertex is colored or a feasible coloring of the graph is not possible. For a graph G = (V, E) the asymmetric game chromatic
number, denoted by γ(G; a, b), is defined as the least integer s such that there is
a winning strategy for Alice in the asymmetric game using s colors. Note that for a = b = 1 it holds γ(G) = γ(G; a, b).
We go a step further by considering the case that the number of the moves varies each time the players take turns. The crucial point is to define the sets of moves as m-tuples ¯a := (a1, ..., am) and ¯b := (b1, ..., bm), where xi vertices
are colored during the ith turn for x ∈ {a, b} and i ∈ {1, ..., m}. We call this new game general asymmetric game and define the general asymmetric game
chro-matic number of a graph G = (V, E), denoted by γg(G; ¯a, ¯b), as the least
in-teger s such that there exists a winning strategy for Alice when the general asymmetric game is played on G using s colors. Obviously, for xi = xj for all
i, j ∈ {1, ..., m} and x ∈ {a, b} it holds
γ(G; a, b) = γg(G; ¯a, ¯b).
This new consideration provides a more general characterization of the ordi-nary asymmetric game since the number of the moves is variable. It is of our interest to investigate γg(G; ¯a, ¯b)for certain distributions of the m-tuples for the
class of cycles, complete multipartite graphs and forests in sections 4.1, 4.2, 4.3, respectively.
Considering a cycle Cn on n vertices, we work out a winning strategy for
Alice for all values aiand bi; in particular, we show that for a1 ≥ dn3e Alice fixes
the coloring of Cnfor n even, after her first turn, if 2 colors are given. For the
case a1 < dn3e we show that there does not exist an optimal strategy for Alice,
i.e., Bob achieves his best case.
Further, we turn our attention to the general asymmetric game chromatic number of complete multipartite graphs. We work out the optimal strategy for Alice and conclude Bob’s worst case for all values ai and bi. Precisely,
winning strategy for Alice for a1 ≥ n, where the coloring of the graph is fixed
after her first turn. A more interesting result is obtained, if we assume that a1 < n. We prove an optimal strategy for Alice and determine the required
number of colors for the cases that Alice or Bob fixes the coloring of the graph, respectively.
For the purpose of investigating the new parameter for the class of forests, we apply techniques by Kierstead, which he worked out for the ordinary asym-metric game in [17]. We give an upper bound for the case ai+1 ≥ bi for all i ∈
{1, ..., m} as well as for the case that there exists one and only one j ∈ {1, ..., m} with aj+1 < bj. Furthermore, we determine lower bounds while working
un-der the assumption that a1 < 2b1and a1 = ... = am where bi is strictly
increas-ing for i ∈ {1, ..., m}. In particular, we prove a winnincreas-ing strategy for Bob and calculate after how many turns he wins the game for some relevant distribu-tions of ¯b = (b1, ..., bm).
Preliminaries
In this chapter we will present some basic definitions and set up notations and terminologies required throughout the thesis. In section 1.2 (Graph Coloring) we will introduce the notions of graph coloring and circular graph coloring, while summarizing without proofs some relevant results. In particular, we will look more closely at vertex-coloring of weighted graphs. Section 1.3 (Games on Graphs) will be devoted to games on graphs. After a brief exposition of the two-person game, we will give the definition of a simplified version of it, called the marking game. Finally, the asymmetric game will be defined. Specific terms and definitions will be introduced in the respective chapters.
1.1 Basic Definitions
Definition 1.1.1. A graph is a pair G = (V, E) of sets such that E ⊆ V
2. V graph
denotes the set of vertices and E denotes the set of edges. We will write V (G) and E(G) instead of V and E if it is not clear from the context.
The figure below shows the well known Petersen graph with vertex set V = {a, b, c, d, e, f, g, h, i, j} and edge set E = {(a, b), (a, e), (a, f), (b, c), (b, g), (c, h), (c, d), (d, i), (d, e), (e, j), (f, h), (f, i), (g, i), (g, j), (h, j)}.
a b c d e f g h i j
Definition 1.1.2. Let G = (V, E) be a graph. Two vertices a, b ∈ V are adjacent
adjacent
if (a, b) is an edge of G, that is, (a, b) ∈ E. Vertex b is also called a neighbor of a and vice versa. We denote the set of all neighbors of b by N(b). A vertex
neighbor
a ∈ V is called incident with an edge e ∈ E if a ∈ e. Two edges e1 6= e2 are
incident
called adjacent if e1 ∩ e2 6= ∅. We call pairwise non-adjacent vertices or edges
independent
independent. Let G = (V, E) be a graph. A set M of independent edges is called
a matching. A perfect matching is a matching M such that every vertex of G is
matching
adjacent to an edge of M.
Throughout the thesis we consider undirected graphs, which means that for every edge (a, b) it holds that (a, b) = (b, a). Edges of the form (x, x) and
multiple edges, where there exist several edges between the same two vertices,
are not allowed. Furthermore, we consider non-empty finite graphs, which means that for the set V it holds that V 6= ∅ and V is finite.
Definition 1.1.3. Let G = (V, E) be a graph and a ∈ V . The degree of a is the
degree
number of the neighbors of a and is denoted by d(a). The maximum degree of G is defined as ∆(G) := max{d(a) | a ∈ V }. A graph is called k-regular if all
k-regular
vertices of G have degree k.
The Petersen graph is 3-regular since each vertex has degree 3.
Definition 1.1.4. A graph G0 = (V0, E0)is called a subgraph of G = (V, E), and
Ga supergraph of G0, if V0 ⊆ V and E0 ⊆ E. In this case we write G0 ⊆ G. If
G0 ⊆ G and G0 contains all edges (a, b) ∈ E with a, b ∈ V0, then G0 is called
induced
Definition 1.1.5. A graph G = (V, E) is calledconnected if for every partition connected
of its vertex set into two non-empty sets X and Y there is an edge with one end in X and one end in Y ; otherwise, the graph is disconnected. Let G0 ⊂ G
be a maximal connected subgraph of G. Then G0 is called a component of G. component
1.2 Graph Coloring
Let S = {1, ..., k} for k ∈ N. A vertex-coloring of a graph G = (V, E) is a map- vertex-coloring
ping c : V → S such that c(u) 6= c(v) whenever (u, v) ∈ E. The elements of
Sare the available colors. The chromatic number χ(G) is the smallest integer k χ(G)
such there is an S with |S| = k and G can be colored with colors from S.
Example: Below we give a coloring for the Petersen graph, where the numbers
indicate the respective colors of the vertices.
1 2 1 2 3 2 1 3 3 2
Circular Coloring of Graphs
Besides the vertex-coloring of graphs there exists a more general coloring, named r-circular coloring, which provides a natural generalization of the vertex-coloring. The basic difference is that in case of circular coloring we assign open unit length arcs on a circle Crwith circumference r ∈ R+instead of colors. The
object of the circular coloring is to figure out the least circumference r of Cr
such that a feasible coloring of the vertices of the graph G is possible. Hence, the result is not necessary a positive integer. In [22] Zhu defined the r-circular
Definition 1.2.1. Let r ≥ 2 be a real number, and let Crdenote a circle of length
r-circular
coloring r. An r-circular coloring of a graph G = (V, E) is a mapping f which assigns to
each vertex x of G an open unit length arc f(x) of Crsuch that f(x) ∩ f(y) = ∅
whenever (x, y) of G. The circular-chromatic number χc(G) of G is equal to the
χc(G)
infimum of those r for which G has an r-circular coloring. For a graph G the following well know relation holds:
χ(G)− 1 < χc(G)≤ χ(G).
Throughout the thesis we will denote an arc on Crby b
(x1,x2)and
character-ize it by its initial-point x1 and end-point x2 on Cr in the clockwise direction,
where x1, x2 ∈ R+. We define the length of an arc b(x1,x2) by
length of an arc l(b(x1,x2)) := x2− x1 if x2 > x1, r− (x1− x2) else.
In particular, for b(x1,x2)∩ b(y1,y2) =∅ and x1 < y1let
distance between two arcs dist(b(x1,x2), b(y1,y2)) := min(y1− x2), (r− (y2− x1)) for 0 /∈ b(y1,y2) min(x1− y2), (y1− x2) else
be the distance between the arcs b(x1,x2)and b(y1,y2).
Example: Consider the circle C6 below and the arcs b
(2,3) and b(5.5,0.5). Then dist(b(2,3), b(5.5,0.5)) = min{(2 − 0.5), (5.5 − 3)} = 1.5. 0 = 6 1 2 3 4 5 C6
The notion of the circular coloring is based on the idea of the (k, d)-coloring and the corresponding star chromatic number χ∗(G), which was introduced by
Vince in [20]. Bondy and Hell showed in [6] that the infimum of χ∗(G) is
at-tained, such that it can be replaced by the minimum. Moreover, they proved that χ∗(G) is a rational number. In [22], Zhu pointed out two alternate
defi-nitions, namely the r-circular coloring (see definition 1.2.1) and the r-interval
coloring, and showed the equivalence of all three definitions. We will give a
detailed insight of the other two definitions in the appendix (A.1). For our purposes, we will consider the notion of the r-circular coloring of graphs. For simplicity of notation we write circular coloring instead of r-circular coloring if it is clear from the context.
Example: Consider the graph below. While this graph has chromatic number 3,
its circular chromatic number is 2, 5. Below we give a 2, 5-circular coloring of this graph. a e d c b a b c d e 0 1 2
Coloring of Weighted Graphs
Definition 1.2.2. A graph (G, w) = (V, E, w) is calledweighted if there exists a weighted graph
mapping w : V → N which assigns to each vertex x a vertex-weight w(x) > 0. Note that if the mapping w takes the constant value 1, then we write G instead of (G, w).
Definition 1.2.3. Let (G, w) = (V, E, w) be a weighted graph and let S =
vertex-coloring of weighted graphs
{1, ..., k} for k ∈ N. A vertex-coloring of a weighted graph (G, w) = (V, E, w) is a mapping c : V → S such that c(x) ≥ w(x) for all x ∈ V and c(u) 6= c(v) when-ever (u, v) ∈ E. The chromatic number of a weighted graph χ(G, w) is equal to the
χ(G, w)
minimum integer k such that there exists a feasible coloring c : V → {1, ..., k} of (G, w).
Example: Consider the weighted graph (G, w) below. Since there exist two
ad-jacent vertices with vertex-weight 5 at least 6 colors are required for achieving a feasible coloring of G. If we assume that each vertex has weight 1, then 3 colors suffice for coloring G.
(G, w) 1 2 1 5 3 5 2 1 4 2 6 3 5 3
FIGURE: Left: a weighted-coloring where the small-size numbers
indicate the vertex-weights while the big-size numbers stand for the colors. Right: a coloring with w : V → {1}.
(G, 1) 3 2 1 3 1 1 1
The theory of weighted graphs has a lot of applications, in particular in daily life. For example one can model a road traffic as a weighted graph by assigning the circulation of the traffic from road i to road j to the vertex vi,j,
where
• the vertex-weight w(vi,j)describes the required duration of a green phase
• two vertices vi,j and uk,lare connected with each other if an overlapping
of the corresponding green phase is not allowed and causes an accident. The chromatic number of the corresponding graph represents the shortest pe-riod for a complete traffic control system. See the following road intersection and the corresponding graph.
bk,k ak,i ci,k di,i ek,k i k
FIGURE(a): A road intersection with w(bk,k) = w(ek,k) = 4, w(ak,i) = 2and w(ci,k) = w(di,i) = 7.
ak,i ci,k bk,k di,i ek,k 4 4 2 7 7 (G, w)
FIGURE(b): The corresponding graph for the road intersection above,
1.3 Games on Graphs
The Two-Person Game on Graphs
In [3] Bodlaender introduced the two-person game on graphs as follows:
Let G = (V, E) be a graph and C be a given set of colors. Two players Alice
the two-person
game and Bob take turns alternately assigning the vertices of G colors from C, suchthat adjacent vertices get distinct colors. Alice starts and wins the game if all
vertices of G are colored, otherwise, Bob wins. The game chromatic number of G,
γ(G)
denoted by γ(G), is the least cardinality of C, such that there exists a winning strategy for Alice.
Obviously, since Alice and Bob are competitive it holds χ(G) ≤ γ(G) ≤ ∆(G) + 1.
Example: Consider the two-person game on the graph G below, where
obvi-ously χ(G) = 2.
a b
c d
If the two-person game is played on G, 2 colors do not suffice in order to achieve a feasible coloring of G. Assume that 2 colors are given. By the structure of the graph Alice is indifferent which vertex to color first. Thus, without loss of generality assume that she starts the game by coloring vertex a with the color 1. Suppose Bob colors vertex c with the color 2. Since the vertices b and d are adjacent to a and c they cannot be colored neither with 1 nor 2. Thus, Bob wins the game. However, it holds γ(G) = 3.
The Marking Game on Graphs
In [14] Faigle, Kern, Kierstead and Trotter informally dealt with a simplified version of Bodlaender’s two-person game, the marking game on graphs. In [26] Zhu formally introduced it as follows:
Let G = (V, E) be a graph and let t ∈ N be a given integer. Alice and Bob
take turns marking vertices from the shrinking set of unmarked vertices with the marking game
Alice playing first. This results in a linear ordering L of the vertices with x < y if x is marked before y. The orientation GL = (VL, EL)of G = (V, E) with
re-spect to L is defined by EL ={(v, u) | (v, u) ∈ E and v > u in L}. The score of the score
the game is defined by 1+∆+
GL, where ∆
+
GLis the maximum number of marked
neighbors of an unmarked vertex during the game. Alice wins the game if the
score is at most the given integer t. The marking game number colg(G) of G is colg(G)
the least integer t such that Alice has a winning strategy for the marking game played on G.
Suppose colg(G) colors are given. If Alice follows her optimal strategy of
the marking game and colors the vertex by First-Fit which is to be chosen, she wins also the coloring game. Thus, this parameter provides an upper bound for the game chromatic number of a graph such that it holds
γ(G)≤ colg(G).
The Asymmetric Game on Graphs
In [17] Kierstead introduced a modified version of Bodlaender’s two-person game on a graph G that differs from it in the following way:
Let G = (V, E) be a graph. Further let C be a given set of colors and a, b ∈ N. Each time Alice and Bob take turns Alice colors a and Bob b vertices in a row. Alice starts coloring and wins the game if there is a feasible coloring of G with the given set C. Otherwise, Bob wins. Note that if all vertices have been colored, the respective player does not have to complete the turn. Obviously, for a = b = 1 we have the regular two-person game on graphs.
The asymmetric game chromatic number, denoted by γ(G; a, b), is the least car- γ(G; a, b)
dinality of C such that there is a winning strategy for Alice in the asymmetric coloring game with the given set C of colors.
Consider the graph G in the example on page 8. We can conclude that for a > 1 the coloring of the graph is fixed after Alice’s first turn, if she colors
adjacent vertices with 1 and 2 or if she colors alternate vertices with the same color. Thus, it holds γ(G; a, b) = 2.
The Circular Two-Person Game on
Graphs
We gave the definition of the two-person game and the game chromatic num-ber γ(G), which was introduced by Bodlaender in [3], on page 8. In this chapter we intend to generalize Bodlaender’s idea by working out a combination of the two-person game and the circular coloring of graphs (see page 4). We will in-troduce the circular two-person game on graphs and define the circular game
chro-matic number. Afterwards, the new parameter will be investigated for the class
of complete graphs, complete multipartite graphs, complete bipartite graphs minus a perfect matching, cycles, cactuses and planar graphs. As mentioned in the introduction, Zhu and Lin independently also worked on the circular two-person game in [27].
The Circular Game Chromatic Number
Consider the following two-person game: Let G = (V, E) be a graph and let Cr be a circle with circumference r ∈ R+. Two players Alice and Bob take
turns coloring vertices of G from the shrinking set U of uncolored vertices with Alice coloring first. In each move the respective player assigns to any vertex v ∈ U an open unit length arc f(v) such that f(u) and f(v) must not overlap whenever (u, v) ∈ E. Alice wins if every vertex can be colored with an arc on Cr. Otherwise, Bob wins.
The circular game chromatic number γc(G)of a graph G = (V, E) is equal to
the infimum of those r for which there exists a winning strategy for Alice on Cr.
This new game turns out to be a natural generalization of Bodlaender’s two-person game with γc(G) ∈ R+ and γ(G) ∈ N. The basic difference is that
the vertices of G are being assigned arcs instead of positive integers. Our ex-ample below demonstrates that not only the choice of the vertices to color is decisive but also the choice of the arcs to assign on the given Cr.
Example: Consider the coloring of the graph G on vertices {v1, v2, v3, v4} where
(v1, v2), (v2, v3), (v3, v4) ∈ E. By symmetry there are two opportunities for
Al-ice’s first turn; either she colors a vertex from {v1, v4} or {v2, v3}. However,
after Alice’s first turn at least two consecutive vertices remain uncolored. Ob-viously, the worst case occurs, if after Bob’s first turn the two neighbors of an uncolored vertex are colored with arcs of distance 1 − ε for an ε > 0 and ε → 0. Without loss of generality assume that dist(f(v2), f (v4)) = 1− ε. Then
v3 cannot not be assigned to an arc between f(v2)and f(v4) in the clockwise
direction because (v2, v3), (v3, v4)∈ E. G Cr v3 v2 v1 v4 f(v2) f(v4) 1− ε
It is clear that for a graph G the ordinary circular chromatic number χc(G)
is less than or equal to γc(G) because Alice and Bob are competitive. For the
trivial upper bound of the circular game chromatic number we consider the coloring of the vertex v and its neighbors N(v), where v is a vertex with maxi-mum degree ∆(G). Let d(v) = ∆(G) = n with N(v) = {v1, ..., vn} and assume
that all vertices from N(v) are colored by Bob but v is uncolored yet. Without loss of generality assume that the vertices {v1, ..., vn} have been assigned to
arcs on Crin the clockwise direction. If it holds that 0 < dist(f(v
i), f (vi+1)) < 1
and i ∈ {1, ..., n − 1}, then v cannot be colored with an arc between f(vi)and
f (vi+1)for i ∈ {1, ..., n − 1}. Hence, we obtain that
χc(G)≤ γc(G)≤ 1 + ∆(G) + (∆(G) − 1) = 2∆(G).
One could conjecture that 1 + ∆(G) + (∆(G) − 1)(1 − ε) for an ε > 0 is sufficient for coloring N(v) ∪ {v}. Assume that a circle Crwith r = 1 + ∆(G) +
(∆(G)− 1)(1 − ε) is given. If Bob colors {v1, ..., vn} with arcs of distance 1 − nε
with v uncolored yet, then dist(f(vi), f (vi+1)) = 1−nε for i ∈ {1, ..., n − 1} and
dist(f (vn), f (v1)) = 1 + n + (n− 1)(1 − ε)−n + (n− 1)(1 − ε n) = 1 + (n− 1)(1 − ε) − (n − 1)(1 − ε n) = 1 + (n− 1)(1 − ε − 1 + ε n) = 1 + (n− 1)(ε n − ε) < 1.
This implies that v cannot be colored on Cr. Thus, Bob wins.
2.1 The Circular Game Chromatic Number of
Com-plete Graphs
In order to determine the circular game chromatic number of complete graphs we need to consider two cases. For Knbeing the complete graph on n vertices
we will distinguish between n odd and even. It is to be expected that by the structure of the Knboth players are indifferent which vertex to color when they
take turns because none of the assigned arcs are allowed to overlap. However, we will work out strategies for both players how to place the corresponding arcs of the chosen vertices.
Definition 2.1.1. A graph G = (V, E) iscomplete if every two vertices are
K4 K3
K2 K1
Proposition 2.1.2. Let Knbe a complete graph. Then
γc(Kn)≤ n +n2 for n odd, n + n 2 − 1 for n even.
Proof. We work out a strategy for Alice and determine the required
circumfer-ence r on a circle Cr for which Alice wins, if she applies this strategy.
Alice’s strategy: Initially, Alice colors an arbitrary vertex with an arbitrary
arc on Cr. Then throughout the game she proceeds as follows. Let v ∈ V be an
uncolored vertex and {f0, ..., fm−1} be the set of the assigned arcs on Cr in the
clockwise direction.
• Assume that there exist two arcs fi and fi+1 for i ∈ {0, ..., m − 1} such
that dist(fimod m, fi+1 mod m)≥ 1. Then Alice colors v with an arc between
fiand fi+1 such that either dist(fi, f (v)) = 0or dist(fi+1, f (v)) = 0.
• Otherwise, if dist(fimod m, fi+1 mod m) < 1 for all i ∈ {0, ..., m − 1}, Bob
wins the game by the assumption that Knis complete.
Bob’s strategy: Each time Bob takes turn he colors an arbitrary vertex with
an arc of distance 1 − ε for an ε > 0 to an already placed arc.
Let n be odd. Then Alice finishes the game, if possible, because she colored first. Thus, during the game she colors n
2 vertices, while Bob colors n 2
ver-tices. The worst case for Alice occurs, if Bob applies his strategy above, which implies that he destroys n
2 additional arcs. Considering these facts we can
conclude that for n odd it holds that γc(Kn)≤ l n 2 m + 2·j n 2 k = n +j n 2 k .
Let n be even. In this case Bob finishes the game; in particular, throughout the game both color n
2 vertices, respectively. Assume that n − 1 vertices are
colored and it is Bob’s turn. Then due to the respective strategies Alice colored
n
2 and Bob n
2 − 1 vertices, while destroying n
2 − 1 additional arcs. Obviously,
for coloring the last vertex an arc of length 1 suffices such that Bob is forced to color with this arc. Thus, we can conclude that for n even it holds that
γc(Kn)≤ n +
n
2 − 1.
2.2 The Circular Game Chromatic Number of
Cy-cles
In this chapter we restrict our attention to the circular game chromatic number of cycles. We work out a strategy for Bob and prove that γc(G) = 4
indepen-dent of which strategy Alice applies.
Definition 2.2.1. Apath is a graph Pn= (V, E)with
V (Pn) ={x1, x2, ..., xn} and E(Pn) ={(x1, x2), (x2, x3), ..., (xn−1, xn)},
where the vertices xi are distinct. The vertices x1 and xn are called the
end-points of Pn. For x1 = xnthe graph is called a cycle on n vertices and is denoted
by Cn, where
E(Cn) = {(x1, x2), (x1, x2), ..., (xn−1, x1)}.
The number of the edges of a path or a cycle is its length. For x, y ∈ V the
distance between u and v is defined as the number of the edges of the shortest
path between u and v and is denoted by d(u,v).
Example: Consider the path P5with E(Pn) ={(x1, x2), (x2, x3), (x3, x4), (x4, x5)}.
x1 x5 x4 x3 x2 P
Proposition 2.2.2. Let Cn= (V, E)be a cycle. Then γc(Cn) = 4.
Proof. Suppose a circle Cr with circumference r = 4 is given and let V =
{x0, x1, ..., xn−1} and E = {(ximod n, xi+1 mod n) | i ∈ {0, ..., n − 1}}. Since each
vertex has degree 2, during the game each uncolored vertex has at most two colored neighbors. Thus, it is sufficient to consider the coloring of an arbitrary path {xj−1 mod n, xjmod n, xj+1 mod n} for j ∈ {1, ..., n} on Cn. Without loss of
generality assume that Bob has colored the vertices xj−1and xj+1such that
dist(f (xj−1), f (xj+1)) = 1− ε for an ε > 0. This coloring illustrates the worst
case that can occur, since the arc of length 1 − ε between f(xj−1)and f(xj+1)
becomes useless for xj. Nevertheless, since r = 4 there exists an arc between
f (xj+1)and f(xj−1)of length 1 with which xj can be colored. Hence, γc(Cn)≤
4. xj−3 xj+1 xj xj−1 xj−2 ... 0 = 4 1 2 3 f(xj−1) f(xj+1) C4
However, a circle with less circumference is not sufficient. Let r = 4 − ε. Without loss of generality assume that Alice colors xj−1 with the arc b(0,1) in
her first turn. If Bob colors xj+1 with the arc b(2−ε 2,3−
ε
2), a feasible coloring of
xj is not possible anymore because l(b(3−ε2,4−ε)) < 1 and l(b(1,2−ε2)) < 1. Hence,
0 = 4− ε 1 2 3 C4−ε b(0,1) b 2−ε 2,3− ε 2
2.3 The Circular Game Chromatic Number of
Com-plete Multipartite Graphs
It is our purpose to study the circular game chromatic number for complete multipartite graphs. First, we give an upper bound for the circular game chro-matic number of a complete multipartite graph Ks1,...,sn for the case that si ≥ 4
for all i ∈ {1, ..., n}. Afterwards, we assume that there exists at least one inde-pendent set with 3 vertices and prove a modified winning strategy for Alice on a circle with less circumference. Afterwards in section 2.3.1 we turn our attention to complete bipartite graphs minus a perfect matching M and show that γc(Km,m)≤ γc(Km,m− M).
Definition 2.3.1. Let n ≥ 2 be an integer. A graph G = (V, E) is called
n-partite or multin-partite if V admits a partition into n classes S1, ..., Sn such that
every edge has its ends in different classes. Vertices in the same partition are not allowed to be adjacent. We denote a multipartite graph by Ms1,...,sn with
independent sets S1, S2, ..., Sn where si = |Si| for i = 1, ..., n. Instead of a
2-partite graph one usually says bi2-partite graph.
For the case that each two vertices from Si and Sj are adjacent we call the
S1 S2 S3 . . . . . . . . S1 S2
FIGURE: The multipartite graph Ms1,s2,s3 with independent sets {S1, S2, S3}
for s1= s2= 2and s3= 3and the complete bipartite graph
Ks1,s2 with independent sets {S1, S2} for s1≥ s2.
By the structure of a complete multipartite graph, it is clear that once a vertex from an independent set Siis colored, the remaining uncolored vertices
of Si can be colored with the same arc. Then we say that Siis save from Alice’s
view.
Proposition 2.3.2. Let Ks1,...,sn = (V, E)be a complete multipartite graph with si ≥
4for all i ∈ {1, ..., n}. Then it holds γc(Ks1,...,sn)≤ 3n − 2.
Proof. The procedure is to work out a strategy for Alice and to determine the
required circle Cr which guarantees her victory.
Alice’s Strategy: Let C ⊆ V be the set of all colored vertices of Ks1,...,sn. Initially,
Alice colors an arbitrary vertex. Then each time she takes turn she goes ahead as follows. Let {f0, ..., fm} be the set of all assigned arcs on Crin the clockwise
direction. Assume that there exists a j0 such that Sj0 ∩ C = ∅.
• If there exist two arcs fimod mand fi+1 mod msuch that
dist(fimod m, fi+1 mod m) ≥ 1 for i ∈ {0, ..., m}, then Alice colors an
arbi-trary vertex v from Sj0 with an arc between fimod m and fi+1 mod m such
that either dist(fimod m, f (v)) = 0or dist(fi+1 mod m, f (v)) = 0.
• If dist(fimod m, fi+1 mod m) < 1for all i ∈ {0, ..., m}, then by the structure of
a complete multipartite graph Alice looses the game because the vertices of Sj0 cannot be colored with a feasible arc.
Let Sj∩ C 6= ∅ for all j ∈ {1, ..., n}. Then Alice chooses the vertices to color at
random such that vertices from the same independent set are colored with the same arc.
What is left is the worst case scenario that can occur, which is the following: Let {f0, ..., fm} be the set of all assigned arcs on Crand let S0 ={S1, ..., Sk} for
1≤ k ≤ n such that Si∩ C 6= ∅ for all i ∈ {1, ..., k}. Suppose that there exists at
least one independent set Slsuch that Sl∩C = ∅. Otherwise, if Si∩C 6= ∅ for all
i ∈ {1, ..., n}, then by the structure of the graph the coloring of the remaining uncolored vertices is fixed.
• Assume that there exist two arcs fimod m and fi+1 mod m for i ∈ {0, ..., m}
such that dist(fimod m, fi+1 mod m) ≥ 2. Then Bob colors an arbitrary
ver-tex from S0 with an arc of distance 1 − ε to fi
mod m for an ε > 0.
• Assume that dist(fimod m, fi+1 mod m) < 2 for all i ∈ {0, ..., m} and that
there exists one j ∈ {0, ..., m} such that 1 ≤ dist(fjmod m, fj+1 mod m) < 2.
Then Bob colors an arbitrary vertex from S0 with an arc of distance 0 to
fjmod mor to fj+1 mod m.
• Assume that dist(fimod m, fi+1 mod m) < 1for all i ∈ {0, ..., m}. Then Bob
wins the game because the vertices of Slcannot be colored with a feasible
arc.
Obviously, if Bob colored in an independent set that does not contain col-ored vertices, a smaller circle would be sufficient for Alice’s victory. Moreover, assigning the arcs with distance of 1 − ε is clearly the maximum he is able to destroy in each move.
Alice’s strategy and Bob’s worst case strategy imply that n distinct unit length arcs are assigned by Alice and n − 1 by Bob, whereas he additionally destroys n − 1 further arcs. Hence, we get the following calculation:
γc(Ks1,...,sn)≤ n + 2(n − 1) = 3n − 2.
Corollary 2.3.3. The circular game chromatic number for bipartite graphs is 4. We have been working under the assumption that every independent set contains at least 4 vertices. It is worth pointing out that less circumference than 3n−2 suffices if we admit that there exists at least one independent set Si0with
si0 = 3. Assume a circle with circumference r = 3n − 3 is given. Then a slight
change in Alice’s strategy is required in order to achieve a feasible coloring on Crfor r = 3n − 3.
Alice’s modified strategy: Without loss of generality assume that s1 = 3 with
S1 ={v11, v12, v13}. Initially, Alice colors vertex v11 with an arbitrary arc. By his
worst case strategy Bob colors vertex v12 such that dist(f(v11), f (v12)) = 1− ε
for an ε > 0. In contrast to her strategy in proposition 2.3.2 Alice does not go ahead with Si for i ∈ {2, ..., n} but colors the remaining uncolored vertex of S1
such that v13 is colored with the arc f(v11)or f(v12). Since s1 = 3, Bob is forced
to jump into {S2, ..., Sn} and to color an uncolored independent set first, which
would be save from Alice’s view. In particular, the required circumference de-creases by 1. However, they proceed the game using the same strategies as in proposition 2.3.2. This implies that throughout the game Alice and Bob assign n− 1 distinct unit length arcs, respectively, whereas Bob additionally destroys n − 1 further arcs. Hence, we can conclude that a circle with circumference r = (n− 1) + 2(n − 1) = 3n − 3 suffices for achieving a feasible coloring of the graph.
2.3.1 The Circular Game Chromatic Number of Complete
Bi-partite Graphs without a Perfect Matching
One may conjecture that the circular game chromatic number decreases if we reduce the number of the edges of a graph. However, it is well known, that 3 = γ(Km,m) ≤ γ(Km,m − M) = m, where the Km,m is a complete bipartite
graph and M is a perfect matching of Km,m. We found out that it also holds for
the circular version of the game, such that for m ≥ 3 4 = γc(Km,m)≤ γc(Km,m− M).
Due to a corollary of the so-called marriage theorem of Hall, [9], every k-regular bipartite graph contains a perfect matching. Thus, since the Km,m is
m-regular, it contains a perfect matching. In the following we will consider the vertex set V (Km,m) as the disjoint union of the two distinct independent
sets, denoted by A and B, where A = {a1, a2, ..., am} and B = {b1, b2, ..., bm}.
Obviously, M = {(a1, b1), (a2, b2), ..., (am, bm)} is a perfect matching. See the
following figure. a1 a2 am b1 b2 bm · · · · · · Km,m− M a1 a2 am b1 b2 bm · · · · · · Km,m
FIGURE: Km,mand Km,m− M with M = {(a1, b1), (a2, b2), ..., (am, bm)}.
Remark 2.3.4. Obviously, the coloring of the entire graph is trivial, if there exists i0 ∈ {1, ..., m} such that f(ai0) ∩ f(bi0) = ∅ because of the following
consideration: Since (ai0, bj)∈ E(Km,m− M) for i0 6= j, none of the remaining
uncolored vertices from {b1, ..., bm} can be colored with an arc that is allowed
to overlap with f(ai0). Thus, the coloring of the remaining uncolored vertices
{a1, ..., am} is fixed, since they can be colored with f(ai0)The same conclusion
can be drawn for the coloring of {b1, ..., bm}.
Proposition 2.3.5. Let Km,m= (V, E)be a complete bipartite graph and M a perfect
matching of Km,mfor m ≥ 3 and m ∈ N. Then
m < γc(Km,m− M) ≤ m + 2.
Proof. Let M = {(a1, b1), (a2, b2), ..., (am, bm)}. We denote by A = {a1, ..., am}
and B = {b1, ..., bm} the two independent sets. For the purpose of determining
the upper bound, we will work out a strategy for Alice and determine the required circumference for achieving a feasible coloring of the graph. Let U denote the set of uncolored vertices and W the set of colored vertices during the game. Obviously, U = V and W = ∅ at the beginning of the game.
• Initially, Alice colors an arbitrary vertex.
• Assume that there exists an i ∈ {1, ..., m} with xi being colored and yi
being uncolored for x, y ∈ {a, b} and x 6= y. Alice colors yi with an arc
such that f(xi)∩ f(yi) = ∅.
• Assume there does not exist an i ∈ {1, ..., m} with xibeing colored and yi
being uncolored for x, y ∈ {a, b} and x 6= y and let the last vertex colored by Bob be an element of H ∈ {A, B}. Further, assume W 6= V . Then Alice colors vertex z ∈ H with an arc such that
– dist(f(z), f(q)) = 0 for q ∈ W and q ∈ {A, B} \ H, – if possible, f(z) ∩ f(z0)6= ∅ for z0 ∈ H.
According to Alice’s strategy we give Bob’s strategy and explain why it turns out to be the worst case scenario.
Bob’s strategy:
(i) Assume |U| ≥ 4. If Alice colors a vertex xjfor x ∈ {a, b} and j ∈ {1, ...m},
then Bob colors yj for y ∈ {a, b} and y 6= x such that l f(xj)∩ f(yj)
= ε and dist(f(yk), f (yj)) = 1− ε for a yk ∈ W and for an ε > 0 (according to
Alice’s strategy such an f(yk)exists).
(ii) Assume |U| ≤ 3 and let xj be the last vertex colored by Alice for x ∈
{a, b}. Bob colors vertex xj0 such that dist(f(xj0), f (xi)) = 1 − ε for an
xi ∈ W .
Further, we explain why Bob’s strategy illustrated above is the worst case for Alice’s strategy. Suppose Alice colors ai and assume that |U| ≥ 4.
• If Bob colored a vertex other than bi, then by the remark 2.3.4 Alice would
be able to fix the coloring in her next turn by coloring bi such that f(bi)∩
f (ai) =∅ holds.
• If Bob colored bi with an arc f(bi)such that f(bi)∩ f(ai) = ∅, then again
Thus, the obvious worst case is to color bi such that l f(ai)∩ f(bi)
= ε and dist(f (bk), f (bi)) = 1− ε for a bk∈ W . The existence of such a bk is guaranteed
since Alice colors vertices with arcs next to already placed arcs. So far, by the strategies of both players we can conclude that m − 1 unit length arcs suffice to color the first 2m − 4 vertices.
Further assume |U| ≤ 3. Without loss of generality we consider the proce-dure of the game where
• the vertices aiand bi for all i ∈ {1, ..., m − 2} have been colored
• Alice started the game coloring a1and
• {am−1, am, bm−1, bm} are uncolored yet.
Since the number of the already colored vertices is even, Alice is the next one to color. In particular, due to the respective strategies of the players, am−2
is the last colored vertex, if m is even, and bm−2 is the last colored vertex, if m
is odd. Without loss of generality assume that m is even and Bob has colored am−2 in his last turn.
Figures (a) and (b) demonstrate the coloring of {a1, b1, ..., am−2, bm−2}, where
M ={(a1, b1), (a2, b2), ..., (am, bm)}. a1 a2 a3 am−3 am−2 am−1 am b1 b2 b3 bm−3 bm−2 bm−1 bm · · · · · ·
FIGURE (a): The dashed lines indicate the edges in the
matching M = {(a1, b1), (a2, b2), ..., (am, bm)} · · · · · · · · f(b2) f(b1) f(a3) f(a2) f(a1) f(am−3) f(am−4) f(bm−2) f(bm−3) f(am−2) m−2 m−1 m−4 m−3 3 2 1 0
Then due to her strategy, Alice colors am−1. At this point of the game we
can conclude that in the independent set A Bob has colored m−2
2 vertices, while
destroying m−2
2 arcs of length 1 − ε. In B Bob also has colored m−2
2 vertices
but since he started coloring in B (since Alice started coloring a1, Bob colored
b1), he has destroyed m2−2 − 1 arcs of length 1 − ε. Thus, obviously it makes
sense for Bob to color am instead of bm−1. Thus, because of (ii), an additional
circumference of 3 is required for coloring bm−1 and am, while another arc of
length 1 − ε is being destroyed.
Finally, we can conclude that γc(Km,m− M) ≤ (m − 1) + 3 = m + 2.
Further, we show that a circle with circumference m does not suffice to guarantee Alice’s victory. In particular, we will give a winning strategy for Bob. In the following we call a free interval of length k an arc of length k on Cm
on which there is no vertex colored.
• Assume that on the circle Cmthere are at least two free intervals of length
2or 3. Then Bob does the following: If Alice colors xi0 where x ∈ {a, b}
and i0 ∈ {1, ..., m}, then Bob colors yi0 where y ∈ {a, b} and y 6= x with
f (xi0). Clearly, if Bob uses this strategy, at some point of the game there
will exist only one free interval of length 2, respectively 3.
If Alice colors such that there exists a free interval of length l or k where 2 < l < 3, respectively, 1 < k < 2, then she obviously loses the game, since this implies that on the circle Cmthere exists a free interval of length
3− l or 2 − k on which there cannot be colored any vertices. By Bob’s strategy, Alice can win the game if and only if at the end there does not exist any free interval on Cm.
• Assume now that it is Alice’s turn and that there is one free interval of length 2 and the other free intervals have length less than 2. Without loss of generality assume that the vertices a1, a2, ..., am−2and b1, b2, ..., bm−2are
colored. Further, we can assume that Alice colors am−1. Then Bob colors
bm−1such that f(am−1)∩ f(bm−1) = εfor an ε > 0 and f(am−1)6= f(bm−1).
For vertex amthere does not exist a feasible arc anymore and hence Bob
... ... f(a1) f(b1) f(a2) f(b2) f(am−2) f(bm−2) f(am−1) f(bm−1) f(am−3) f(bm−3) m 1 2 m−1 m−2 m−3 m−4
• Assume now that it is Alice’s turn and there is one free interval of length 3 and the other free intervals have length less than 3. Without loss of generality assume that the vertices a1, a2, ..., am−3 and b1, b2, ..., bm−3 are
colored. Further we can assume that Alice colors am−2. Two cases are
possible. Either Alice colors am−2 such that a free interval of length 2 is
left or two free intervals of length 1.
Case 1: Assume Alice colors am−2 such that a free interval of length 2 is
produced. Then Bob colors bm−2 so that f(am−2)∩ f(bm−2) = ε. After
Al-ice’s next move, Bob will obviously have the opportunity to color either bm−1 or bm with b(m−1,m) such that he wins the game because for either
am−1or amthere does not exist a feasible arc.
... ... f(a1) (f b1) f(a2) f(b2) f(am−3) f(bm−3) f(am−2) f(bm−2) f(am−4) f(bm−4) m 1 2 m−1 m−2 m−3 m−4 m−5
Case 2: Assume Alice colors am−2 such that two free intervals of length
1 are produced. Then Bob colors bm−2 so that f(am−2)∩ f(bm−2) = ε.
After Alice’s next move, Bob will obviously have the opportunity to color either bm−1 or bm with b(m−1,m) or b(m−3,m−2) such that he wins the game
because for either am−1 or am there does not exist a feasible arc.
... ... a1 b1 f(a2) f(b2) f(am−3) f(bm−3) f(am−2) f(bm−2) f(am−4) f(bm−4) m 1 2 m−1 m−2 m−3 m−4 m−5
2.4 Circular Game Chromatic Number of Planar
Graphs
Planar graphs received huge interest in the graph theoretic community. The definition is easy to explain. A graph is called planar iff it can be drawn on a plane without edge crossing. A lot of strong theorems have been proved on this topic. The most famous is the theorem of Kuratowski, which proves that a graph is planar if and only if it does not contain a K5 or a K3,3. A further
essentially conjecture is the Four Color Conjecture, which claims that every pla-nar graph can be colored with 4 colors. So far there does not exist an ordipla-nary mathematical proof. However, 1977 K. Appel and W. Haken presented a proof of the four color theorem using computer sciences. Concerning the game chro-matic number, there does not exist a precise result for planar graphs. However, a lot of efforts have been made to find sharp upper bounds. The best known is 17worked out by Zhu in [25]. Our approach is to give an upper bound for the circular game chromatic number of planar graphs.
Kierstead bounded in [11] the marking game number of a graph G (see page 8) which restricts the game chromatic number in terms of a parameter r(G), the rank of a graph G. In particular, he worked out a winning strategy for Alice for the marking game, the so called activation strategy, and proved that the game chromatic number of planar graphs is at most 18. Our approach is to extend the marking game and to introduce the circular marking game on graphs. In addition, we intend to determine the circular game chromatic number of planar graphs using techniques by Kierstead.
2.4.1 The Activation Strategy for the Circular Game
• We denote by Π(G) the set of linear orderings on the vertices of G. • For a linear ordering L we obtain the orientation GL of G with respect toLby setting EL={(v, u) : {v, u} ∈ E and v > u in L}.
• Let VG+L(u) = {v ∈ V |v < u} and V
−
GL(u) = {v ∈ V |v > u} with V
+ GL[u] =
• For a vertex u ∈ V (G) we denote the outneighborhood of u in GL by
N+
GL(u) and the inneighborhood of u in GL by N
−
GL(u) with N
+ GL[u] =
NG+L(u)∪ {u} and N
−
GL[u] = N
−
GL(u)∪ {u}. The various degrees of u are
denoted by d+ GL(u) = |N + GL(u)| and d − GL(u) = |N −
GL(u)|. The maximum
outdegree of GLis denoted by ∆+GL and the maximum indegree by ∆−GL.
x1 x2 x3 x4 x5 x6 x7 x8 FIGURE: GLwith EL={(x2, x1), (x3, x2), (x4, x3), (x5, x4), (x6, x3), (x7, x6), (x8, x7), (x8, x1), (x8, x5)}
The Circular Marking Game
The circular marking game is played on a finite graph G by Alice and Bob with Alice playing first. In each move the players take turns choosing vertices from the shrinking set U ∈ V (G) of unchosen vertices. This results in a linear order-ing L of the vertices of G with x < y iff x is chosen before y. Notice L ∈ Π(G) where Π(G) is the set of linear orderings on V (G). The c-score of the game is equal to 1 + ∆+
GL + (∆
+
GL − 1) = 2∆
+
GL. Alice wins, if the c-score is at most a
given integer t; otherwise, Bob wins.
The circular marking game number colc
g(G) of a graph G = (V, E) is the least
integer t such that Alice has a winning strategy for the circular marking game, that is 2∆+
GL ≤ t.
Remark 2.4.1. Note that the circular marking game differs from the ordinary marking game in terms of the score of the game. While the score of the ordi-nary marking game is ∆+
GL+ 1, where ∆
+
GLis defined as the maximum number
of marked neighbors of an unmarked vertex, it turns out to be the worst case for the coloring game. The idea behind, is that the neighbors N(v) of a vertex v
get distinct colors. For bounding the circular game chromatic number we have to go a step further and to consider the positions of the corresponding arcs of N (v) on the cycle Cr, while assuming that Bob colored them. Thus, another
arc of length ∆+
GL− 1 is required, if the worst case occurs.
Proposition 2.4.2. For a graph G = (V, E) it holds γc(G)≤ colgc(G).
Proof. Suppose we have a circle with circumference colc
g(G). If Alice follows
her optimal strategy of the circular marking game and colors the vertex by First-Fit, which is to be chosen, she wins also the circular coloring game.
The Activation Strategy S(L, G) and the Rank of G
Kierstead introduced the activation strategy which restricts ∆+
GLif Alice applies
this strategy. For this purpose, he defined the so called rank of a graph G. For the sake of completeness we will give below this strategy.
Fix a graph G and a linear ordering L ∈ Π(G). Let A ⊂ V be the set of
active vertices with A := ∅ at the beginning. Furthermore let U denote the set
of unchosen vertices. Alice starts the game by activating the least vertex from L and chooses it. After Bob has chosen any vertex b from U Alice does the following: Strategy S(L, G) • x := b; • while x /∈ A do A := A ∪ {x}; s(x) := minLN+[x]∩ (U ∪ {b}); x:= s(x) od; • if x 6= b then choose x
else y := minLU; if y 6∈ A then A := A ∪ {y} fi; choose y
The Rank of a Graph
Let A, B ⊂ V (GL). A matching M is a matching from A to B if M saturates A
and B \A contains a cover of M. As mentioned above ∆+
GLis bounded in terms
of the following parameters. For u ∈ V (G) the matching number m(u, L, G) of u with respect to L is defined as the size of the largest set Z ⊂ N−[u] such
that there exists a partition {X, Y } of Z and there exist matchings M from X ⊂ N−[u]to V+(u)and N from Y ⊂ N−(u)to V+[u]. The rank r(L, G) of G
with respect to L and rank r(G) of G are defined by r(u, L, G) := d+GL(u) + m(u, L, G)
r(L, G) := max
u∈V r(u, L, G)
r(G) := min
L∈Π(G)r(L, G)
For our case we shall extend this definition and define the circular rank of a graph G:
The Circular Rank of a Graph
rc(u, L, G) := 2(d+GL(u) + m(u, L, G))− 1
rc(L, G) := max u∈V r c(u, L, G) rc(G) := min L∈Π(G)r c(L, G)
The next proposition shows the relation between the circular game marking number and the circular rank of a graph G.
Proposition 2.4.3. For any graph G = (V, E) and ordering L ∈ Π(G), if Alice uses the strategy S(L, G) in order to play the circular marking game on G, then the c-score will be at most 1 + rc(L, G). In particular, colc
g(G) ≤ 1 + rc(G).
Proof. Suppose that Alice applies strategy S(L, G) for the circular marking
game on G. Since every vertex chosen by Bob immediately becomes active and any vertex chosen by Alice is already active, it remains to show that at any time t any unchosen vertex u is adjacent to at most d+
GL(u) + m(u, L, G)active
vertices instead of chosen vertices, that is |N(u) ∩ A| ≤ d+
It is obvious that |N(u) ∩ A| ≤ d+
GL(u) +|N
−(u)∩ A|, and since Kierstead
proved in [11] that |N−(u)∩ A| ≤ m(u, L, G), we can conclude that
|N(u) ∩ A| + |N(u) ∩ A| − 1 ≤
d+GL(u) +|N−(u)∩ A| + d+GL(u) +|N−(u)∩ A| − 1 ≤
d+GL(u) + m(u, L, G) + d+GL(u) + m(u, L, G)− 1 = rc(u, L, G)
In proposition 2.4.3 we proved that colc
g(G)is bounded by the circular rank.
In order to determine the circular game chromatic number of a graph G or a class of graphs one can determine the circular rank by the proposition 2.4.2. In the following we will use this result to bound γcfor planar graphs.
The Circular Game Chromatic Number of Planar Graphs Corollary 2.4.4. If G is a planar graph, then colc
g(G) ≤ 34.
Proof. Let G = (V, E) be a planar graph. Kierstead proved in [11] that at any
time for an unchosen vertex u ∈ V (G) it holds: d+GL(u) + m(u, L, G)≤ 17.
Using this result, we found out that for the vertex u ∈ V (G) rc(u, L, G)
≤ 33
holds. It follows for the circular marking game number of a planar graph G that
colcg(G) ≤ 34.
In this manner, according to [11], we are able to estimate easily the circular game chromatic number for trees.
Definition 2.4.5. A graph that does not contain any cycles is called aforest. A
connected forest is called a tree and is denoted by T = (V, E). Remark 2.4.6. If T = (V, E) is a tree, then colc
g(G)≤ 6.
Proof. Let L be an ordering of V so that |N+
L(v)| ≤ 1 for every v ∈ V (G)
(since T is a tree such a linear ordering can be easily found). Then obviously colc
g(G)≤ 6.
2.5 The Circular Game Chromatic Number of
Cac-tuses
Definition 2.5.1. A graph K = (V, E) is acactus if any two cycles of K have at
most one common vertex.
The aim of this section is to determine the circular game chromatic number for the class of cactuses by using techniques of Sidorowicz in [19]. Sidorowicz showed that the game chromatic number for the class of cactuses is 5. First, she proved that 5 is an upper bound, using lemma 2.5.3 proved by Zhu in [26], by showing that colg(C) ≤ 5 for C being the class of cactuses. Further, she proved
that there exists a cactus for which Alice has no winning strategy if four colors are given. In particular, Sidorowicz showed the following lemma.
Lemma 2.5.2. If K = (V, E) is a cactus, then there is a matching M such that K−M is an acyclic graph.
Note that an acyclic graph is a graph which does not contain any cycles. Hence, we can decompose a cactus K = (V, E) into a forest F and a match-ing M with K = F ∪ M. Usmatch-ing the lemma below Sidorowicz was able to give an upper bound.
Lemma 2.5.3. Suppose G = (V, E) and E = E1 ∪ E2. Let G1 = (V, E1) and
G2 = (V, E2). Then colg(G)≤ colg(G1) + ∆(G2).
(V, E)that colg(K) ≤ 4 + 1 = 5 and hence γ(K) ≤ 5. As mentioned above she
proved further the existence of a cactus with game chromatic number greater or equal to 5 and finally conclude that the game chromatic number for the class of cactuses is equal to 5. We will use these techniques for determining the circular game chromatic number for the class of cactuses. Further we will call a vertex v ∈ V (G) for a graph G pendant if d(v) = 1. First, we give the circular version of Zhu’s lemma.
Lemma 2.5.4. Suppose G = (V, E) and E = E1 ∪ E2. Let G1 = (V, E1) and
G2 = (V, E2). Then colgc(G) ≤ colcg(G1) + 2∆(G2).
Proof. Suppose t = colc
g(G1) + 2∆(G2). If Alice applies her optimal strategy of
G1 to play the marking game on G, then she wins:
Assume Alice applies her optimal strategy of G1 and assume that L ∈
Π(G1)is a linear ordering of V which is produced through the game. For this
linear order L we can conclude:
∆+G 1L + 1 + (∆ + G1L − 1) ≤ col c g(G1)⇒ ∆+G1L + 1 + (∆+G1L − 1) + 2∆(G2)≤ colgc(G1) + 2∆(G2). Obviously colc g(G)≤ ∆+G1L + 1 + (∆ + G1L − 1) + 2∆(G2)
holds and hence
colgc(G)≤ col c
g(G1) + 2∆(G2).
According to lemma 2.5.4 we can conclude for the class of cactuses:
Corollary 2.5.5. Let C be the class of cactuses. Then colc
g(C) ≤ 8.
Further, we prove that there exists a cactus with circular game chromatic number greater or equal to 5. In particular, we will consider the cactus below (by Sidorowicz [19]) which we will denote by K0.
! ! " " # # $ $ % % & & ' ' ( ( ) ) * * + + w1 w2 w3 v3 v2 v1 v4 K0
Lemma 2.5.6. For the cactus K0 it holds γc(K0)≥ 5.
Proof. Suppose a circle with circumference 5 − ε for an ε > 0 is given. Assume
Alice starts the game coloring v1 with α. Without loss of generality let α =
b(0,1). Then Bob colors v3 with b(2−ε 4,3−
ε
4). This forces Alice to color v2 in her
next turn. She must color it with an arc between 3 − ε
4 and 5 − ε. However,
Alice colors v2, Bob easily can color a pendant vertex of w1 or w2 such that a
circle of circumference 5 − ε does not suffice for coloring w1or w2, respectively.
If Alice starts coloring another vertex, then Bob colors a vertex of distance 2
The Two-Person Game on Weighted
Graphs
In this chapter we will introduce a modification of Bodlaender’s two-person game by taking vertex-weights into account. Thus, throughout the chapter we will consider a weighted graph (G, w) such that to the vertices of (G, w) are assigned positive integers, the so called vertex-weights.
While for coloring a graph without vertex-weights the coloring mapping c : V → N must satisfy the condition c(u) 6= c(v) for (u, v) ∈ E, for a feasible coloring of a weighted graph an additional condition for c must hold:
c(v)≥ w(v),
where w(v) is the vertex-weight of the vertex v ∈ V . The difficulty of the new concept is that for the strategies of the respective players, besides the structure of the graph also the given distribution of the vertex-weights is decisive. We wish to investigate the two-person game on weighted graphs and to discuss the game chromatic number γ(G, w) for the class of complete graphs, com-plete multipartite graphs, cycles, trees and planar graphs. For the purpose of bounding γ(G, w), we will extend the marking game for our case and define the weighted marking game number, denoted by colw
g(G), which turns out to be an
upper bound of γ(G, w). For determining the weighted marking game number we will modify in section 3.6 Kierstead’s activation strategy, which we already pointed out on page 28.
